Is it possible to edit bone positions on the posed (not rest) stance?
If I understand correctly, you want to change your rest pose, but permanently, so that when you go into Edit Mode, the bone positions do not revert to any other position, because they've been truly changed from the position they were in originally, making the new position the new default rest pose, even in Edit Mode. There is a way to do this, and it's not difficult, but the steps are somewhat counter-intutive.
With your desired pose in place, go to Object Mode and select your character mesh. Go to the Modifiers tab in the Properties Panel, and locate your existing Armature modifier. Duplicate it. You may notice that the mesh jumps, but don't worry. For the moment, it's having the tweaks it's receiving from armature modifiers doubled because there are now two of them. Thankfully, this will be a very short-lived issue.
The next thing you want to do is apply the instance of these armature modifiers which is higher in the stack, and keep the other one.
Now go into Pose Mode and select all your bones. Press Ctrl + a and then from the Apply menu, select Apply Pose as Rest Pose.
You're done. When you go into Edit Mode, you will see that the armature really is in the new pose.
Hope this helps.